Method and apparatus for determining the yaw angle of a satellite

ABSTRACT

The method according to the invention allows to determine the yaw angle of a satellite from the reading of two different sensors measuring the roll and/or pitch angles, provided that the reference point of the two sensors are not identical. A description is given basically for geostationary satellites but the method can be applied directly to satellites which are stationary with respect to any star. The method can be employed for circular and non-circular orbits.

[0001] The invention relates to a method and an apparatus fordetermining the yaw angle of a satellite.

[0002] In order to effectively control satellites, especiallygeostationary satellites, the exact orientation of the satellite has tobe known. Apart from other values, the roll angle, the pitch angle andthe yaw angle, which will described in greater detail further below,have to be measured or estimated. Most geostationary, so called“three-axis stabilized” satellites are provided with sensors that allowto measure and actively control the roll and pitch angle. In this case,no sensor is provided for measuring the yaw angle. The yaw angle isusually estimated and controlled by means of the roll/yaw coupling thatoccurs throughout the orbit when an angular momentum bias is present onthe spacecraft, for instance when a momentum wheel is spinning inside.Since this coupling is actually very light, a proper yaw angleestimation and correction takes several hours. Although thesespacecrafts also are usually equipped with rate measurement assemblieslike gyroscopes with a fast measurement response around all three axes,the latter are only used for autonomous attitude measurements over shortperiods of time for instance when the attitude is expected to bedisturbed like during station keeping maneuvers. The reason is that theintegrated angle tends to drift away due to the presence of an inherentbias in the measured rates. In addition, when the rate measurementdevices are gyroscopes the risk of mechanical wear leads the operatorsto turn them off whenever their use is not absolutely required.

[0003] Even though these types of satellites usually control their yawangle very acceptably without measuring it, there are cases when a fastyaw measurement is highly desirable like after an unexpected attitudedisturbance or when not enough angular momentum bias is present to allowsufficient coupling between the roll and yaw angles, for instance whenthe momentum wheel has failed. A daily monitoring of the yaw angleprofile is also useful to evaluate the health of the attitude controlsystem.

[0004] Against this background the technical problem to be solved is toprovide a method and an apparatus for determining the yaw angle of asatellite on the basis of sensor measurement signals readily availableat the satellite, i.e. without the requirement of additional sensors.

[0005] This problem is solved by a method according to the claims 1 to4, by a method according to the claims 5 to 11, by an apparatusaccording to the claims 14 to 17 and by an apparatus according to theclaims 18 to 24.

[0006] A very advantageous aspect of the invention is the fact that theinvention needs no estimation schemes which introduce a considerabledelay in computing the yaw angle. Rather the invention makes use of adirect measurement of the yaw angle by means of sensors already presenton the satellite. This makes it possible to provide a fast yawmeasurement avoiding to collect hours of data before be able to infer agood yaw estimation.

[0007] The method is not based on a model matching estimation scheme asmentioned hereabove, but on a direct measurement of the yaw angle purelybased on the geometry of the sensors. In other words, this method doesnot require long data collection periods, but only needs one measurementon two sensors to infer a yaw angle. This is particularly of interestwhen the spacecraft undergoes some unexpected attitude disturbance. Inthis case, if a model-matching estimation scheme is used, since themodel would not fit the observation, the whole data collection should berestarted after the disturbance for proper yaw estimation. With themethod according to the invention, only one measurement, at some pointin time, of two offset sensors would be needed before, during or afterthe disturbance to infer a valid yaw measurement. Furthermore, thismethod does not suffer any observability problem. Hence, the method isfast and reliable and is perfectly suited for real time or historicalmonitoring of yaw pointing.

[0008] Another important aspect of the invention is to monitor theperformances of all different subsystems of a satellite for supervisionand error recovery.

[0009] This monitoring sometimes highlights deficiencies of somesubsystems. Upon deficiency detection, corrective operations areundertaken to either enhance the performance or prevent further damages.Monitoring the proper functioning of the yaw pointing control is part ofthis overall monitoring task. A poor or unexpected yaw pointing is notonly liable to affect the satellite's mission (poor broadcasting, poorimages, . . . ) but is also a clue of poor attitude control, possiblydue to on-board hardware or software faults. An usual way to monitor theyaw pointing is to record the roll pointing profile over several hoursor days. Indeed, due to the lack of direct yaw sensor mounted on-boardof the spacecraft, many of these spacecrafts control or estimate theiryaw angle based on the orbital coupling between roll and yaw angles. Byrecording the roll angle profile over a sufficient amount of time, theground station can infer the yaw angle profile over the same time span.This is done by applying some numerical estimation scheme to the rollangle data, using a numerical model of the spacecraft's dynamics andkinematics. For instance, a least square fitting can be worked out inthat workframe. These estimation methods are based on the adjustment ofthe model's parameters to match the way the roll profile has varied overthe period of observation. However, not only these methods require longobservation durations for accurate yaw estimation, due to low couplingbetween roll and yaw, they also require very good modeling of theenvironmental disturbance torques for proper modeling of the spacecraftdynamics. If, during the roll angle data collection period, anythingunexpected happens to the spacecraft attitude that is not taken intoaccount into the numerical model of the spacecraft, like an externaldisturbance, the resulting estimated yaw angle could be way off the realyaw angle. In this case, the roll angle data collection must berestarted after the unexpected event which further delays the yawestimation. In addition, depending on the available sensors, the exposedyaw angle estimation method often suffers an observability problem.Indeed, the aforementioned environmental torque modeling is usuallyachieved through a limited fourier series development of the torquesaround the yaw and roll axes. If the spacecraft only disposes of a rollangle and a pitch angle measurement device, the constant term of thefourier series of the yaw environmental torque is not distinguishablefrom the yaw angle itself. Therefore, in this case, the yaw is notestimated as such, but the estimation outputs the sum of the yaw angleand the influence of the constant term of the yaw environmental torque.In this configuration, it is impossible to verify the magnitude of theparasitical effect of the torque's constant term. If this effect islarge, it will significantly bias the whole yaw estimation.

[0010] Hence, the method according to the invention can be implementedin real time and allows the operator not only to verify the magnitude ofthe yaw angle at some point in time, but also to track the profile ofthe yaw angle throughout hours, days or years. It for instance allowscomparisons of daily yaw profiles to make sure there is no attitudecontrol performance degradation or unexpected variation in yaw angle. Itcould also be looked at if there is any clue of yaw transient (e.g. poorbroadcasting).

[0011] A further aspect of the present invention deals with thecalibration of the yaw angle measurement which is performed as mentionedabove.

[0012] The theory assumes perfectly stable sensors whose boresights arealways pointing in the same direction with respect to the satellite'scoordinate frame. However, real attitude sensors are not perfect. Theirreading and pointing are for instance quite sensitive to thermalvariation or aging. For instance, due to the daily rotation of ageostationary spacecraft's body, the structure on which a sensor ismounted follows a daily distortion cycle due to cycling sun exposure.Another source of spurious error can be a initial misalignment of thesensor on the spacecraft body.

[0013] Hence, applying the yaw measurement as mentioned above, a furtherproblem occurs in calibrating this yaw measurement also readilyavailable at the satellite without a requirement of additional sensors.

[0014] This problem is solved by a method according to claim 12, by amethod according to claim 13, by an apparatus according to claim 25 andby an apparatus according to claim 26.

[0015] The method according to the invention allows to measure the yawangle from the reading of two different sensors measuring the rolland/or pitch angles, provided that the reference point of the twosensors are not identical. The description is given basically forgeostationary satellites but the invention can be applied directly tosatellites which are stationary with respect to any star. The method canalso be employed for non circular orbits. The method assumes that theorbit of the spacecraft is known at any time.

[0016] In the following, the method and the apparatus according to theinvention and its principles will be explained in greater detail withreference to the drawings of which

[0017]FIG. 1 shows a view of an orbiting satellite in an earth orbit forillustrating a reference coordinate system.

[0018]FIG. 2 shows a view of the earth from orbiting satellite includinggeometric indicators for explaining roll and pitch angles with andwithout the presence of a yaw angle.

[0019]FIG. 3 shows a geometric relation for explaining the principles ofthe method according to the invention.

[0020]FIG. 4 shows a view of the earth from an orbiting satelliteincluding geometric indicators for explaining the principles of themethod according to the invention.

[0021] In order to define the position and orientation of a satellite,there is a need for the definition of a reference frame, i.e. a tripletof axes virtually attached to the body of the satellite, the latterbeing considered as infinitely rigid.

[0022] The following discussion will be limited to geostationarysatellites. In this case, the location of the center of the frame(orbit) is less involved in the following development. It should benoted that in any case the orbit only affects the location of thereference points of the sensors on earth. So if the orbit is notgeostationary, the location of the reference points on the earthdeterministically vary throughout this orbit.

[0023] The three axes are named roll, pitch and yaw and are oriented asshown in FIG. 1, i.e. roll: along the velocity vector of the satellite(tangential to the orbit) pitch: perpendicular to the orbit plane of thesatellite, directed towards south; yaw: nominally towards the center ofthe earth so as to close the tri-orthogonal right-hand oriented frame.

[0024] The orientation of the satellite, further called “attitude” canbe specified in terms of rotations about these three axes so that onemay speak of roll, pitch and yaw angles. Roll, pitch and yaw angles areactually “Euler angles”. This means that they represent the orientationof a body with respect to a reference orientation by consecutiverotations around the corresponding axes. The final orientation dependson the order of rotations. In other words, if the yaw/pitch/roll orderof rotations is picked, then starting from the reference frame of FIG.1, the final attitude is reached by first rotating the frame around theyaw axis, then rotating the frame around the new pitch axis, then aroundthe new roll axis. The final orientation would not be the same if therotations were performed, with the same magnitude, but in a differentorder. However, if the magnitude of the rotations are small enough, thefinal attitude becomes almost independent of the order of rotations.This approximation is often applied when the attitude errors of aproperly controlled geostationary satellite are considered, since itspointing errors are very small.

[0025]FIG. 2 shows the earth seen from a satellite, as well as thedirections P and R in which pitch and roll errors will move a beam sentby the satellite on the earth. The solid lines show the directions alongwhich roll and pitch errors would be measured if there were no yawerror, the dotted lines give the same information but when some positiveyaw error is present.

[0026] As an example and as shown in FIG. 2, points G and N locatedistinct reference points of two sensors provided on-board thesatellite. The reference point of each sensor is defined as the locationit points to when the errors it reads are zero.

[0027] As an example, two sensors used on-board the ASTRA satellites areconsidered in the following, but different kinds of sensors may be usedas well. One sensor is an optical infrared earth sensor assembly (ESA)with the subnadir point N (center of the earth) as its reference point.The other sensor is a beacon sensor with the ground station G as itsreference point. Each sensor issues roll and pitch angle attitude errorsdefining the difference between the direction it points to, its“boresight”, and its reference point (identified by points G and N). Thesatellite transmits the telemetry values of the measured roll and pitchangles of both sensors to the ground station which records them forfurther processing and/or analysis. The roll and pitch errors of atleast one of the sensors are also sent to the on-board processor forroll and pitch control.

[0028] It should be noted again that the method explained below extendsto any kind of pair of sensors measuring roll and pitch angles or twolinear combination of these angles, as long as the reference points Gand N of the two sensors are different. In addition, the method can alsoreadily be extended to a point N not being on the center of the earth.

[0029] The roll and pitch angles are often represented like planarcoordinates as in FIG. 2. However, since these angles are actually Eulerangles, this representation is only valid for small angles.

[0030] According to the definitions above, roll and pitch errors with azero yaw error would show up as shown in FIG. 3, where R1, P1 and R2, P2are the errors measured by the two sensors in roll and pitch. Of course,the length of the segment joining the boresights of the two sensors whenthere is an off-pointing (an attitude error) is the same as when thereis no error (line joining points N and G).

[0031] If a yaw error y exists, as shown in FIG. 4, the direction inwhich roll and pitch errors are measured would be canted by y. In thissituation, the readings of R1, P1 and R2, P2 are not equal anymorebetween the two sensors. This means information about the yaw angle iscaptured in the difference between the readings of the two sensors.

[0032] In order to retain only the yaw information, and since R1 and P1are assumed to be small angles, which means that a planar representationcan be applied, the actual roll and pitch errors can be eliminated bygeometrically translating R2 and P2 respectively by R1 and P1 ending upon point G′ as shown in FIG. 5. When using the difference between thereading of the roll error on one hand and between the readings of thepitch error on the other hand two angle Delta_p and Delta_r can bedefined:

Delta_p=P 2-P 1  (1)

Delta_r=R 2-R 1  (2)

[0033] In other words, the problem is now reduced to a pure yaw angle,resulting in the second sensor to read Delta_p and Delta_r namely thecoordinates of point G′, and the first sensor to read zero.

[0034] Although not really necessary, it can be assumed that Delta_p andDelta_r are small angles. This slightly simplifies the considerationsbelow. This assumption has been verified for all ASTRA spacecrafts sincea yaw angle of the order of a degree leads to Delta_p and Delta_r whichare about one tenth thereof.

[0035] In order to find the yaw angle provoking a Delta_p and Delta_r,that yaw angle has to be determined that would rotate point G to pointG′, i.e. that would make the second sensor to read the errors Delta_pand Delta_r. Knowing that point G is defined by the consecutive pitch(azimuth) and roll (elevation) rotations that bring the center of theearth to the reference point of the second sensor and point G′ isdefined as the pitch and roll rotations that bring the center of theearth, as seen from the spacecraft, to a location where the secondsensor reads Delta_p and Delta_r.

[0036] The following equations represent the rotations of a vectorpointing from the spacecraft to the center of the earth: on one side, apitch rotation (pitch_GP) followed by a roll rotation (roll_GP) with noyaw rotation which bring the center of the earth to G′ and, on the otherside, a pitch rotation (pitch_G) followed by a roll rotation (roll_G)which bring the center of the earth to G, and finally followed by a yawrotation that brings point G to point G′. $\begin{matrix}{\begin{bmatrix}1 & 0 & 0 \\0 & {\cos \left( {{roll\_}{GP}} \right)} & {- {\sin \left( {{roll\_}{GP}} \right)}} \\0 & {\sin\left( {{roll\_}{GP}} \right)} & {\cos\left( {{roll\_}{GP}} \right)}\end{bmatrix} \cdot {\quad{{\begin{bmatrix}{\cos\left( {{pitch\_}{GP}} \right)} & 0 & {\sin\left( {{pitch\_}{GP}} \right)} \\0 & 1 & 0 \\{- {\sin\left( {{pitch\_}{GP}} \right)}} & 0 & {\cos\left( {{pitch\_}{GP}} \right)}\end{bmatrix} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}} = {\quad{\begin{bmatrix}{\cos ({yaw})} & {- {\sin ({yaw})}} & 0 \\{\sin ({yaw})} & {\cos ({yaw})} & 0 \\0 & 0 & 1\end{bmatrix} \cdot \begin{bmatrix}1 & 0 & 0 \\0 & {\cos\left( {{roll\_}G} \right)} & {- {\sin\left( {{roll\_}G} \right)}} \\0 & {\sin \left( {{roll\_}G} \right)} & {\cos\left( {{roll\_}G} \right)}\end{bmatrix} \cdot {\quad{\begin{bmatrix}{\cos\left( {{pitch\_}G} \right)} & 0 & {\sin\left( {{pitch\_}G} \right)} \\0 & 1 & 0 \\{- {\sin\left( {{pitch\_}G} \right)}} & 0 & {\cos\left( {{pitch\_}G} \right)}\end{bmatrix} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}}}}}}}} & (3)\end{matrix}$

[0037] The above expression defines a system of three equations withonly “yaw” as an unknown. For the ASTRA spacecrafts, the variablespitch_G and roll_G are actually the azimuth and elevation of the groundstation (Betzdorf, Luxembourg) as seen from the spacecraft, while sinceDelta_p and Delta_r are small the angles pitch_GP and roll_GP can bedefined as follows:

pitch_GP=pitch_G+Delta_p  (4)

roll_GP=roll_G+Delta_r.  (5)

[0038] Developing system (3), it yields:

sin(pitch_GP)=cos(yaw)*sin(pitch_G)+sin(yaw)*sin(roll_G)*cos(pitch_G)  (6)

−sin(roll_GP)cos(pitch_GP)=sin(yaw)*sin(pitch_G)−cos(yaw)*sin(roll_G)*cos(pitch_G)  (7)

cos(roll_GP)*cos(pitch_GP)=cos(roll_G)*cos(pitch_G)  (8)

[0039] With

A=sin(pitch_GP)  (9)

B=−sin(roll_GP)*cos(pitch_GP)  (10)

C=sin(pitch_G) and  (11)

D=sin(roll_G)*cos(pitch_G)  (12)

[0040] and by eliminating cos(yaw) between equations (6) and (7), theabove equations (6), (7) and (8) can be reduced to

sin(yaw)=(B*D−A*C)/(B*B+C*C)  (13)

[0041] and by substitution,

cos(yaw)=(A*B+D*C)/(B*B+C*C)  (14)

[0042] Since the yaw angle is usually small (a couple of degrees at moston ASTRA spacecrafts), equation (13) is often enough to determine theyaw angle and its sign. Therefore, most of the time, only one equationis sufficient. However, equation (8) and (14) can provide additionalinformation to refine the yaw angle.

[0043] The yaw angle as measured this way has already been shown to fitthe integration of yaw gyro rates, for instance. It will be aninvaluable source of attitude information, especially in cases where theknowledge of the yaw angle becomes critical.

[0044] The above described method is based on measurement values fromtwo sensors as mostly used in geostationary satellites, for example theASTRA satellites. However, the principle of the invention can also beused even if measurement values of only two single sensors, for exampletwo single roll sensors or two single pitch sensors are available sincetwo independent equations exist. In the following, it will be assumedthat all variables, except <<yaw>> and <<roll_GP>> are known. In otherwords, only the pitch angle is measured.

[0045] However, the location of G is still known in roll and pitch. Wecan use eq. (6) to determine <<yaw>>. Indeed, only two of the abovegiven equations (6), (7) and (8) are independent. The combination ofequations (6), (7) and (8) reflects only that the length of the unitvector (0 0 1) is conserved throughout the successive rotations. Hence,two independent equations exist so that two variables could be unknown.It should be noted that the third equation could still be used to helpsolving the problem. Previously, it had been demonstrated that theproblem was solved easily if all variables, except <<yaw>> were known.In the following, it will be assumed that all variables, except <<yaw>>and <<roll_GP>> are known. In other words, only the pitch angle ismeasured. However, the location of G is still known in roll and pitch.We can use eq. (6) to determine <<yaw>>.

[0046] Developing system (3), it yields:

−cos(yaw)*sin(pitch_G)=sin(yaw)*sin(roll_G)*cos(pitch_G)−sin(pitch_GP)  (15)

[0047] and squaring both sides of equation (15) leads to:

[1−sin²(yaw)]*sin²(pitch_G)=sin²(yaw)*sin²(roll_G)*cos²(pitch_G)+sin²(pitch_GP)−2*sin(yaw)*sin(roll_G)*cos(pitch_G)*sin(pitch_GP)  (16)

[0048] and finally to a quadratic equation in sin(yaw):

sin²(yaw)*[sin²(pitch_G)−sin²(roll_G)*cos²(pitch_G)]=sin(yaw)*[2*sin(roll_G)*cos(pitch_G)*sin(pitch_GP)]=sin²(pitch_G)−sin²(pitch_GP)=0  (17)

[0049] which can easily be solved to determine the yaw angle as follows:

[0050] Let:

E=(sin(pitch_G)*sin(pitch_G))−(sin(roll_G)*cos(pitch_G)*sin(roll_G)*cos(pitch_G))

F=2*sin(roll_G)*cos(pitch_G)*sin(pitch_GP);

G=sin(pitch_G)*sin(pitch_G)−sin(pitch_GP)*sin(pitch_GP);

sin(yaw)=(−F±sqrt(F*F−4*E*G))/(2*E);

[0051] Then perform the arcsin on sin(yaw) if abs(sin(yaw))<=1.Practically, pick yaw closest to zero if several solutions areavailable.

[0052] Another simplifying is possible by solving the yaw equations with2 sensor measure measurements on the basis of roll channels only.

[0053] Taking back equations (7) and (8):

−sin(roll_GP)cos(pitch_GP)=sin(yaw)*sin(pitch_G)−cos(yaw)*sin(roll_G)*cos(pitch_G)

cos(roll_GP)*cos(pitch_GP)=cos(roll_G)*cos(pitch_G)

[0054] ″cos(pitch_GP) can be eliminated from these two equations,giving:

−sin(roll_GP)*cos(roll_G)*cos(pitch_G)/cos(roll_GP)=sin(yaw)*sin(pitch_G)−cos(yaw)*sin(roll_G)*cos(pitch_G)

sin(roll_GP)*cos(roll_G)*cos(pitch_G)/cos(roll_GP)+sin(yaw)*sin(pitch_G)=cos(yaw)*sin(roll_G)*cos(pitch_G)

[0055] let

M=sin(roll_GP)*cos(roll_G)*cos(pitch_G)/cos(roll_GP)

[0056] squaring both sides of this equation gives:

M^ 2+sin^ 2(yaw)*sin^ 2(pitch_G)+2*sin(yaw)*sin (pitch_(13 G))*M=(1−sin^2(yaw))*sin^ 2(roll_G)*cos^ 2(pitch_G)

[0057] and finally, again ends up in a quadradtic equation in sin(yaw)where everything is known except the yaw

sin^ 2(yaw)*(sin^ 2(pitch_G)+sin^ 2(roll_G)*cos^ 2(pitch_G))+

sin(yaw)*2*sin(pitch_G)*M+

M^ 2−sin^ 2(roll_G)*cos^ 2(pitch_G)

=0

[0058] Again, letting

H=(sin^ 2(pitch_G)+sin^ 2(roll_G)*cos^ 2(pitch_G))

K=2*sin(pitch_G)*M

[0059] and

L=M^ 2−sin^ 2(roll_G)*cos^ 2(pitch_G)

[0060] yields the following equation

sin(yaw)=(−K±sqrt(K*K−4*H*L))/(2*H);

[0061] Then perform the arcsin on sin(yaw) if abs(sin(yaw))<=1.Practically, pick yaw closest to zero if several solutions areavailable.

[0062] These methods with single channels from both sensors allow tocollect less data than a method with both roll and pitch data from bothsensors. They allow to keep computing the yaw even if roll or pitch dataare not available for some reason.

[0063] Then, equations (6) and (7) can be used to determine the deltaroll angle:

sin(roll_GP)=−[sin(yaw)*sin(pitch_G)−cos(yaw)*sin(pitch_G)−cos(yaw)*sin(roll_)*cos(pitch_(—G)]/cos(pitch)_GP)  (18)

cos(roll_GP)=cos(roll_G)*cos(pitch_G)/cos(pitch_GP)  (19)

[0064] Obviously, the same kind of demonstration can be done if<<roll_GP>> is known and <<pitch_GP>> is not.

[0065] Thus it has been demonstrated that the yaw angle can bedetermined with two offset sensors, each one reading only one attitudeangle, as long as the reference point of each sensor is known. It shouldbe noted that this method determines <<yaw>> and <<roll_GP>> or<<pitch_GP>>, the latter ones being deduced from delta angles betweenthe two sensor readings and not representing the real roll or pitchangle. Therefore, if the roll and pitch angles are needed for control,this method cannot determine them in addition to the yaw angle, but atleast one of the sensors has to measure both roll and pitch. However, asdemonstrated here above, in this particular case, both roll and pitchangles do not have to be measured to determine the yaw angle.

[0066] The following general approach will show that actually threemeasures are necessary to infer the roll, pitch and yaw angles. In otherwords, one of the sensor has to measure both roll and pitch and one ofthe sensor could measure either roll or pitch to allow the fulldetermination of the attitude, i.e. roll, pitch and yaw.

[0067] Assume the same equations, but this time without any assumptionon the value of the angles or on the reference point of the sensors(except being distinct):

[0068] For the first sensor: ${\begin{bmatrix}1 & 0 & 0 \\0 & {\cos ({RM1})} & {- {\sin ({RM1})}} \\0 & {\sin ({RM1})} & {\cos ({RM1})}\end{bmatrix} \cdot \begin{bmatrix}{\cos ({PM1})} & 0 & {\sin ({PM1})} \\0 & 1 & 0 \\{- {\sin ({PM1})}} & 0 & {\cos ({PM1})}\end{bmatrix} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}} = {\quad{{\begin{bmatrix}{\cos (Y)} & {- {\sin (Y)}} & 0 \\{\sin (Y)} & {\cos (Y)} & 0 \\0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 \\0 & {\cos (R)} & {- {\sin (R)}} \\0 & {\sin (R)} & {\cos (R)}\end{bmatrix}} \cdot {\quad{\begin{bmatrix}{\cos (P)} & 0 & {\sin (P)} \\0 & 1 & 0 \\{- {\sin (P)}} & 0 & {\cos (P)}\end{bmatrix} \cdot {\quad{\begin{bmatrix}1 & 0 & 0 \\0 & {\cos ({RR1})} & {- {\sin ({RR1})}} \\0 & {\sin ({RR1})} & {\cos ({RR1})}\end{bmatrix} \cdot {\quad{{\begin{bmatrix}{\cos ({PR1})} & 0 & {\sin ({PR1})} \\0 & 1 & 0 \\{- {\sin ({PR1})}} & 0 & {\cos ({PR1})}\end{bmatrix} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}}{For}\quad {the}\quad {second}\quad {{sensor}:\quad {\left\lbrack \quad \begin{matrix}1 & 0 & 0 \\0 & {\cos ({RM2})} & {- {\sin ({RM2})}} \\0 & {\sin ({RM2})} & {\cos ({RM2})}\end{matrix} \right\rbrack \cdot \left\lbrack \quad \begin{matrix}{\cos ({PM2})} & 0 & {\sin ({PM2})} \\0 & 1 & 0 \\{- {\sin ({PM2})}} & 0 & {\cos ({PM2})}\end{matrix} \right\rbrack \cdot {\quad{\left\lbrack \quad \begin{matrix}0 \\0 \\1\end{matrix} \right\rbrack = {\quad{\left\lbrack \quad \begin{matrix}{\cos (Y)} & {- {\sin (Y)}} & 0 \\{\sin (Y)} & {\cos (Y)} & 0 \\0 & 0 & 1\end{matrix} \right\rbrack \cdot \left\lbrack \quad \begin{matrix}1 & 0 & 0 \\0 & {\cos (R)} & {- {\sin (R)}} \\0 & {\sin (R)} & {\cos (R)}\end{matrix} \right\rbrack \cdot {\quad{\left\lbrack \quad \begin{matrix}{\cos (P)} & 0 & {\sin (P)} \\0 & 1 & 0 \\{- {\sin (P)}} & 0 & {\cos (P)}\end{matrix} \right\rbrack \cdot \quad {\quad{\left\lbrack \quad \begin{matrix}1 & 0 & 0 \\{\quad 0} & {\cos ({RR2})} & {- {\sin ({RR2})}} \\0 & {\sin ({RR2})} & {\cos ({RR2})}\end{matrix}\quad \right\rbrack \quad \cdot {\quad{\left\lbrack \quad \begin{matrix}{\cos ({PR2})} & 0 & {\sin ({PR2})} \\0 & 1 & 0 \\{- {\sin ({PR2})}} & 0 & {\cos ({PR2})}\end{matrix} \right\rbrack \cdot \left\lbrack \quad \begin{matrix}0 \\0 \\1\end{matrix} \right\rbrack}}}}}}}}}}}}}}}}}}}}$

P, R, Y = pitch, roll, yaw angle errors. RRx, PRx = roll and pitchangles of the sensor's reference point (Depending on the orbit and onthe direction to which the sensor points in the spacecraft coordinates;azimuth and elevation of the reference point of sensor ‘x’ as seen fromthe spacecraft). RMx, PMx = measured roll and pitch angles (from centerof a star or from the center of earth).

[0069] All previous cases presented earlier were actually approximationsor particular cases of this last case. Multiplying all these matricestogether for both sensors, will provide three equations for eachsensors. Again, only two of these equations are independent for eachsensor. It provides four independent equations and therefore allows 4unknowns. This time, the small angle approximation (<=10°) on the pitchand roll angle errors <<P>> and <<R>> is not applied and hence onecannot deduct the readings of both sensors to get around thedetermination of <<R>> and <<P>>.

[0070] Therefore, by default, three unknown variables exist: <<R>>,<<P>> and <<Y>>, what allows one more unknown. This means that forinstance one of the following measure could be unknown: RM1, RM2, PM1 orPM2. Hence, in general, in order to fully determine the attitude, onesensor has to measure both roll and pitch and one sensor can measureonly the roll or the pitch angle. This is actually not surprising sinceone has a static geometrical problem with three independent unknownvariables and hence one has to measure three independent values.

[0071] Obviously the general system given here above can be developed bymultiplying all matrices, but the resolution of this system is actuallymore difficult than the one that was presented earlier. The easiest waywould be to solve the corresponding system numerically. A numericalresolution could be developed in another report if needed.

[0072] The need of calibration will be explained according to FIG. 4 bymeans of a distortion. This distortion is equivalent to a spuriousoffset or variation of point G or of point N on FIG. 4. This leads tospurious roll and pitch measurements on the affected sensor, which inturn leads to spurious yaw measurement with the method.

[0073] Most part of these spurious sensor errors are constant errors(alignment errors) and one orbit varying (daily for geostationaryspacecrafts).

[0074] In order to get rid of the constant and of the daily variation,the method must be calibrated. For calibration a quiet orbit is needed,which means a full orbit revolution without attitude disturbance andwithout station keeping moves, without unusual external disturbances andwithout unusual activity affecting the satellite's attitude. The quietorbit according to the invention is used as a reference orbit. Theoutput of all angle sensors are then used for yaw measurement and arecollected over the reference orbit which is 24 hours of data collectionfor geostationary satellites.

[0075] According to a first aspect of calibration, a yaw profile iscomputed from the collected data.

[0076] Yaw is measured with the method according to the invention withthe collected data over the quiet orbit. Note that sampling time can beadjusted to match typical rate of variation of the yaw angle. When yawangle has been computed over the quiet orbit, the method is ready forbeing used at any point in time. The yaw angle is computed at thedesired time as explained above. The yaw computed at the same orbitlocation on the reference orbit is then subtracted from this computedyaw and the result gives the calibrated yaw angle.

[0077] To summarize,

[0078] 1. pick a quiet orbit;

[0079] 2. collect necessary data as explained above to compute yawprofile over this quiet orbit=Yreference(t) where t=time in referenceorbit;

[0080] 3. measure the necessary data at the time yaw measurement isdesired and compute a yaw angle=Yaw_unbiased at time t0, where t0=timeof day of measurement;

[0081] 4. subtract point 2. result from point 3. result:

[0082] Yaw_calibrated at time t0=(Yaw_unbiased at time t0)−Yreference(t0).

[0083] This calibration method reduces the measuring of the yaw relativeto the yaw that was measured at the same point on the reference orbit.It assumes that the yaw profile observed on the reference orbit was areal yaw pointing offset and variation.

[0084] According to a second aspect of calibration, a sensor profile isrecorded for each angle sensor on the satellite.

[0085] This calibration consists of again picking a quiet orbit andrecording necessary roll or pitch data to compute yaw on this orbit.Then, at the time yaw measure is desired, the ground station measuresthe set of roll or pitch data required to compute yaw. Then, deduct rollor pitch data collected at the same location on the reference orbit fromrespective roll or pitch data at the time yaw measurement is desired.Then, compute yaw based on calibrated set of roll or pitch data.

[0086] To summarize,

[0087] 1. pick a quiet orbit;

[0088] 2. collect necessary data as explained above to compute yawprofile over this quiet orbit. For instance Roll_reference_sensor1(t)and Roll_reference_sensor2(t) where t=time in reference orbit;

[0089] 3. collect necessary data as explained above at the time yawmeasurement is desired. For instance Roll_sensor1 at time t0 andRoll_sensor2 at time t0;

[0090] 4. subtract reference data from respective data at time yawmeasurement is desired:

[0091] * Roll_sensor1 at t0 calibrated=

[0092] (Roll_sensor1 at time t0)—Roll_reference_sensor1(t0)

[0093] * Roll_sensor2 at time t0 calibrated=

[0094] (Roll_sensor2 at time t0)—Roll_reference_sensor2(t0)

[0095] 5. compute calibrated yaw angle with “Roll_sensor1 at time t0calibrated” and “Roll_sensor2 at time t0 calibrated”.

[0096] This calibration method assumes that the yaw remained equal tozero throughout the reference orbit. It assumes that the yaw that wouldhave been computed with the sensor errors on this reference orbit wouldonly have been due to spurious errors on the sensors, like the onescoming from thermal distortion or alignment errors.

1. Method for determining the yaw angle of a satellite, wherein a fixedcartesian coordinate system is defined with regard to the satellite,having a first axis which is directed substantially tangential to theorbit of the satellite and defines a roll angle, having a second axiswhich is directed substantially perpendicular to the orbit plane of thesatellite and defines a pitch angle, and having a third axis which isdirected substantially radial to the orbit of the satellite and definesa yaw angle, comprising the steps of: providing a first angle sensor anda second angle sensor having different reference points and measuringthe same angle about a measurement axis which is anti-parallel to thethird axis, evaluating a first angle and a second angle about themeasurement axis on the basis of the angle sensors, and determining theyaw angle on the basis of the first angle and the second angle. 2.Method according to claim 1, wherein the measurement axis is the firstaxis.
 3. Method according to claim 1, wherein the measurement axis isthe second axis.
 4. Method according to claim 3, wherein the satelliteis a geo-stationary satellite, wherein the first angle sensor has itsreference point at the center of earth (N) and the second angle sensorhas its reference point at a ground station (G) located apart from thecenter of earth.
 5. Method for determining the yaw angle of a satellite,wherein a fixed cartesian coordinate system is defined with regard tothe satellite, having a first axis which is directed substantiallytangential to the orbit of the satellite and defines a roll angle,having a second axis which is directed substantially perpendicular tothe orbit plane of the satellite and defines a pitch angle, and having athird axis which is directed substantially radial to the orbit of thesatellite and defines a yaw angle, comprising the steps of: providing afirst angle sensor and a set of angle sensors including a second andthird angle sensor, wherein the first, second and third angle sensorsmeasure angles about different measurement axes which are anti-parallelto the third axis and wherein for the first angle sensor and for the setof angle sensors different reference points are assigned, evaluatingmeasurement angles on the basis of the angle sensors, and determiningthe yaw angle on the basis of the measurement angles.
 6. Methodaccording to claim 5, wherein the set of angle sensors comprises a firstroll angle sensor (R1) and a first pitch angle sensor (P1) both relatedto a first reference point (G), and wherein the first angle sensorcomprises a second roll angle sensor (R2) or a second pitch angle sensor(P2), related to a second reference point (N) which is different fromthe first reference point (G).
 7. Method according to claim 6, whereinfor the first sensor calculations are based on equations representableby the following expression: ${\begin{bmatrix}1 & 0 & 0 \\0 & {\cos ({RM1})} & {- {\sin ({RM1})}} \\0 & {\sin ({RM1})} & {\cos ({RM1})}\end{bmatrix} \cdot \begin{bmatrix}{\cos ({PM1})} & 0 & {\sin ({PM1})} \\0 & 1 & 0 \\{- {\sin ({PM1})}} & 0 & {\cos ({PM1})}\end{bmatrix} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}} = {\quad{{\begin{bmatrix}{\cos (Y)} & {- {\sin (Y)}} & 0 \\{\sin (Y)} & {\cos (Y)} & 0 \\0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 \\0 & {\cos (R)} & {- {\sin (R)}} \\0 & {\sin (R)} & {\cos (R)}\end{bmatrix}} \cdot \begin{bmatrix}{\cos (P)} & 0 & {\sin (P)} \\0 & 1 & 0 \\{- {\sin (P)}} & 0 & {\cos (P)}\end{bmatrix} \cdot {\quad{\begin{bmatrix}1 & 0 & 0 \\0 & {\cos ({RR1})} & {- {\sin ({RR1})}} \\0 & {\sin ({RR1})} & {\cos ({RR1})}\end{bmatrix} \cdot \begin{bmatrix}{\cos ({PR1})} & 0 & {\sin ({PR1})} \\0 & 1 & 0 \\{- {\sin ({PR1})}} & 0 & {\cos ({PR1})}\end{bmatrix} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}}}}}$

and wherein for the second sensor calculations are based on equationsrepresentable by the following expression: ${\begin{bmatrix}1 & 0 & 0 \\0 & {\cos ({RM2})} & {- {\sin ({RM2})}} \\0 & {\sin ({RM2})} & {\cos ({RM2})}\end{bmatrix} \cdot \begin{bmatrix}{\cos ({PM2})} & 0 & {\sin ({PM2})} \\0 & 1 & 0 \\{- {\sin ({PM2})}} & 0 & {\cos ({PM2})}\end{bmatrix} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}} = {\quad{\begin{bmatrix}{\cos (Y)} & {- {\sin (Y)}} & 0 \\{\sin (Y)} & {\cos (Y)} & 0 \\0 & 0 & 1\end{bmatrix} \cdot \begin{bmatrix}1 & 0 & 0 \\0 & {\cos (R)} & {- {\sin (R)}} \\0 & {\sin (R)} & {\cos (R)}\end{bmatrix} \cdot \begin{bmatrix}{\cos (P)} & 0 & {\sin (P)} \\0 & 1 & 0 \\{- {\sin (P)}} & 0 & {\cos (P)}\end{bmatrix} \cdot {\quad{\begin{bmatrix}1 & 0 & 0 \\0 & {\cos ({RR2})} & {- {\sin ({RR2})}} \\0 & {\sin ({RR2})} & {\cos ({RR2})}\end{bmatrix} \cdot \begin{bmatrix}{\cos ({PR2})} & 0 & {\sin ({PR2})} \\0 & 1 & 0 \\{- {\sin ({PR2})}} & 0 & {\cos ({PR2})}\end{bmatrix} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}}}}}$

with P, R, Y pitch, roll yaw angle errors, RRx, PRx roll and pitchangles of the sensor's reference point, and RMx, PMx measured roll andpitch angles (from center of a star or from the center of earth),

wherein one of the measurements RMx or PMx could be unknown.
 8. Methodaccording to claim 7, wherein the first and the second reference points(G, N) are on the earth.
 9. Method according to claim 8, wherein thefirst and/or second reference points (G, N) are on a star.
 10. Methodaccording to claim 9, wherein the satellite is a geo-stationarysatellite.
 11. Method according to claim 5, wherein the attitude of thesatellite is determined by using the determined yaw angle.
 12. Methodfor determining a calibrated value for a yaw angle of a satellite,comprising the steps of: storing a yaw angle profile over a quiet orbit,determining a sample yaw angle at a calibration time (t0), determining acalibrated value by subtracting the sample yaw angle with a yaw angleprofile value having a time shift in the yaw angle profile correspondingto the calibration time (t0).
 13. Method for determining a calibratedvalue for a yaw angle of a satellite, storing for each angle sensor onboard of the satellite a profile over a quiet orbit, determining asample value for each of the angle sensors at a calibration time (t0),determining calibrated values for each angle sensor by subtracting thesample values with a angle sensor profile value having a time shift inthe respective angle sensor profile corresponding to the calibrationtime (t0), determining a calibrated value for a yaw angle withcalibrated values for each angle sensor.
 14. Apparatus for determiningthe yaw angle of a satellite, wherein a fixed cartesian coordinatesystem is defined with regard to the satellite, having a first axiswhich is directed substantially tangential to the orbit of the satelliteand defines a roll angle, having a second axis which is directedsubstantially perpendicular to the orbit plane of the satellite anddefines a pitch angle, and having a third axis which is directedsubstantially radial to the orbit of the satellite and defines a yawangle, comprising: a first angle sensor and a second angle sensor havingdifferent reference points, means for measuring the same angle about ameasurement axis which is anti-parallel to the third axis, means forevaluating a first angle and a second angle about the measurement axison the basis of the angle sensors, and means for determining the yawangle on the basis of the first angle and the second angle. 15.Apparatus according to claim 14, wherein the measurement axis is thefirst axis.
 16. Apparatus according to claim 14, wherein the measurementaxis is the second axis.
 17. Apparatus according to claim 16, whereinthe satellite is a geo-stationary satellite, wherein the first anglesensor has its reference point at the center of earth (N) and the secondangle sensor has its reference point at a ground station (G) locatedapart from the center of earth.
 18. Apparatus for determining the yaw ofa satellite, wherein a fixed cartesian coordinate system is defined withregard to the satellite, having a first axis which is directedsubstantially tangential to the orbit of the satellite and defines aroll angle, having a second axis which is directed substantiallyperpendicular to the orbit plane of the satellite and defines a pitchangle, and having a third axis which is directed substantially radial tothe orbit of the satellite and defines a yaw angle, comprising: a firstangle sensor and a set of angle sensors including a second and thirdangle sensor, means for measuring by the first, second and third anglesensors angles about different measurement axes which are anti-parallelto the third axis, wherein for the first angle sensor and for the set ofangle sensors different reference points are assigned, means forevaluating measurement angles on the basis of the angle sensors, andmeans for determining the yaw angle on the basis of the measurementangles.
 19. Apparatus according to claim 18, wherein the set of anglesensors comprises a first roll angle sensor (R1) and a first pitch anglesensor (P1) both related to a first reference point (G), and wherein thefirst angle sensor comprises a second roll angle sensor (R2) or a secondpitch angle sensor (P2), related to a second reference point (N) whichis different from the first reference point (G).
 20. Apparatus accordingto claim 19, wherein for the first sensor calculations are based onequations representable by the following expression: ${\begin{bmatrix}1 & 0 & 0 \\0 & {\cos ({RM1})} & {- {\sin ({RM1})}} \\0 & {\sin ({RM1})} & {\cos ({RM1})}\end{bmatrix} \cdot \begin{bmatrix}{\cos ({PM1})} & 0 & {\sin ({PM1})} \\0 & 1 & 0 \\{- {\sin ({PM1})}} & 0 & {\cos ({PM1})}\end{bmatrix} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}} = {\quad{{\begin{bmatrix}{\cos (Y)} & {- {\sin (Y)}} & 0 \\{\sin (Y)} & {\cos (Y)} & 0 \\0 & 0 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & 0 \\0 & {\cos (R)} & {- {\sin (R)}} \\0 & {\sin (R)} & {\cos (R)}\end{bmatrix}} \cdot \begin{bmatrix}{\cos (P)} & 0 & {\sin (P)} \\0 & 1 & 0 \\{- {\sin (P)}} & 0 & {\cos (P)}\end{bmatrix} \cdot {\quad{\begin{bmatrix}1 & 0 & 0 \\0 & {\cos ({RR1})} & {- {\sin ({RR1})}} \\0 & {\sin ({RR1})} & {\cos ({RR1})}\end{bmatrix} \cdot \begin{bmatrix}{\cos ({PR1})} & 0 & {\sin ({PR1})} \\0 & 1 & 0 \\{- {\sin ({PR1})}} & 0 & {\cos ({PR1})}\end{bmatrix} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}}}}}$

and wherein for the second sensor calculations are based on equationsrepresentable by the following expression: ${\begin{bmatrix}1 & 0 & 0 \\0 & {\cos ({RM2})} & {- {\sin ({RM2})}} \\0 & {\sin ({RM2})} & {\cos ({RM2})}\end{bmatrix} \cdot \begin{bmatrix}{\cos ({PM2})} & 0 & {\sin ({PM2})} \\0 & 1 & 0 \\{- {\sin ({PM2})}} & 0 & {\cos ({PM2})}\end{bmatrix} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}} = {\quad{\begin{bmatrix}{\cos (Y)} & {- {\sin (Y)}} & 0 \\{\sin (Y)} & {\cos (Y)} & 0 \\0 & 0 & 1\end{bmatrix} \cdot \begin{bmatrix}1 & 0 & 0 \\0 & {\cos (R)} & {- {\sin (R)}} \\0 & {\sin (R)} & {\cos (R)}\end{bmatrix} \cdot \begin{bmatrix}{\cos (P)} & 0 & {\sin (P)} \\0 & 1 & 0 \\{- {\sin (P)}} & 0 & {\cos (P)}\end{bmatrix} \cdot {\quad{\begin{bmatrix}1 & 0 & 0 \\0 & {\cos ({RR2})} & {- {\sin ({RR2})}} \\0 & {\sin ({RR2})} & {\cos ({RR2})}\end{bmatrix} \cdot \begin{bmatrix}{\cos ({PR2})} & 0 & {\sin ({PR2})} \\0 & 1 & 0 \\{- {\sin ({PR2})}} & 0 & {\cos ({PR2})}\end{bmatrix} \cdot \begin{bmatrix}0 \\0 \\1\end{bmatrix}}}}}$

with P, R, Y pitch, roll yaw angle errors, RRx, PRx roll and pitchangles of the sensor's reference point, and RMx, PMx measured roll andpitch angles (from center of a star or from the center of earth),

wherein one of the measurements RMx or PMx could be unknown. 21.Apparatus according to claim 20, wherein the first and the secondreference points (G, N) are on the earth.
 22. Apparatus according toclaim 21, wherein the first and/or second reference points (G, N) are ona star.
 23. Apparatus according to claim 21, wherein the satellite is ageo-stationary satellite.
 24. Apparatus according to claim 23, whereincomprising means for determining the attitude of a satellite. 25.Apparatus for determining a calibrated value for a yaw angle of asatellite, comprising: means for storing a yaw angle profile over aquiet orbit, means for determining a sample yaw angle at a calibrationtime (t0), means for determining a calibrated value by subtracting thesample yaw angle with a yaw angle profile value having a time shift inthe yaw angle profile corresponding to the calibration time (t0). 26.Apparatus for determining a calibrated value for a yaw angle of asatellite, means for storing for each angle sensor on board of thesatellite a profile over a quiet orbit, means for determining a samplevalue for each of the angle sensors at a calibration time (t0), meansfor determining calibrated values for each angle sensor by subtractingthe sample values with a angle sensor profile value having a time shiftin the respective angle sensor profile corresponding to the calibrationtime (t0), means for determining a calibrated value for a yaw angle withcalibrated values for each angle sensor.